To be fair to the architects of the Millennium Bridge, the structural engineers used a vehicle bridge model, not one for pedestrians.

Multiple comments seem to imply the Millenium Bridge has some really egregious design flaw. So I'm curious: what is it?

It has a resonance mode at 2Hz, now successfully damped. Bridge designers know to look out for resonances at those frequencies, because pedestrians tend to identify and accentuate them by walking, especially in large numbers immediately after opening. The resonance mode wasn't identified during design because unusually it is in the horizontal plane - most bridge designs have very slow resonances in that plane (sway) but the millennium bridge is a side mounted suspension bridge, with the pylons alongside the roadway.

This video explains the problem - it is worth watching the entire video. If you are pressed for time jump to 15:00 mark https://www.youtube.com/watch?v=t-_VPRCtiUg

In 99% of cases yes, but the millennium bridge was a special case.